### Bessel Functions and General SolutionsThis page is dedicated to explaining the use of Bessel functions to determine the general solution to a differential equation. #### Problem StatementYou are given the following differential equation:\[ x^2 y'' + x y' + \left( x^2 - 256 \right) y = 0 \]The task is to express a general solution for this equation using Bessel functions of either the first or second kind.#### Solution Approach1. **Identify the Standard Form:** The given differential equation is of the standard form for Bessel's equation: \[ x^2 y'' + x y' + (x^2 - \nu^2) y = 0 \] Comparing this with the given equation: \[ x^2 y'' + x y' + \left( x^2 - 256 \right) y = 0 \] We identify that $\nu^2 = 256$, which gives $\nu = 16$.2. **Expression Using Bessel Functions:** The general solution to the Bessel's differential equation is given by: \[ y(x) = c_1 J_\nu(x) + c_2 Y_\nu(x) \] where $J_\nu(x)$ and $Y_\nu(x)$ are the Bessel functions of the first and second kinds, respectively, and $c_1$ and $c_2$ are arbitrary constants.Therefore, for our specific case, where $\nu = 16$,\[ y(x) = c_1 J_{16}(x) + c_2 Y_{16}(x) \]#### Final Solution\[ y(x) = \boxed{ c_1 J_{16}(x) + c_2 Y_{16}(x) } \]Here $ c_1 $ and $ c_2 $ are constants determined by initial conditions or boundary values specific to the problem.

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Express a general solution to the given equation using Bessel functions of either the first or second kind. x²y" + xy' + (x² – 256) y = 0 Express any Bessel functions using the notation J,(x) and Y,u(x). y(x) =D

Express a general solution to the given equation using Bessel functions of either the first or second kind. x²y" + xy' + (x² – 256) y = 0 Express any Bessel functions using the notation J,(x) and Y,u(x). y(x) =D

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

### Bessel Functions and General Solutions

This page is dedicated to explaining the use of Bessel functions to determine the general solution to a differential equation.

#### Problem Statement

You are given the following differential equation:

\[ x^2 y'' + x y' + \left( x^2 - 256 \right) y = 0 \]

The task is to express a general solution for this equation using Bessel functions of either the first or second kind.

#### Solution Approach

1. **Identify the Standard Form:**
The given differential equation is of the standard form for Bessel's equation:
\[ x^2 y'' + x y' + (x^2 - \nu^2) y = 0 \]

Comparing this with the given equation:
\[ x^2 y'' + x y' + \left( x^2 - 256 \right) y = 0 \]

We identify that $\nu^2 = 256$, which gives $\nu = 16$.

2. **Expression Using Bessel Functions:**
The general solution to the Bessel's differential equation is given by:
\[ y(x) = c_1 J_\nu(x) + c_2 Y_\nu(x) \]

where $J_\nu(x)$ and $Y_\nu(x)$ are the Bessel functions of the first and second kinds, respectively, and $c_1$ and $c_2$ are arbitrary constants.

Therefore, for our specific case, where $\nu = 16$,

\[ y(x) = c_1 J_{16}(x) + c_2 Y_{16}(x) \]

#### Final Solution

\[ y(x) = \boxed{ c_1 J_{16}(x) + c_2 Y_{16}(x) } \]

Here $ c_1 $ and $ c_2 $ are constants determined by initial conditions or boundary values specific to the problem.

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